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May 25, 2014

Spans and the Categorified Heisenberg Algebra (Part 2)

Posted by John Baez

Last summer I gave a little course on something I really like: Jeffrey Morton and Jamie Vicary’s work on the ‘categorified Heisenberg algebra’ discovered by Mikhail Khovanov. It ties together combinatorics and the math of quantum theory in a fascinating way… related to nice old ideas, but revealing a new layer of structure. I blogged about that course here, with links to slides and references.

The last two weeks I was in Paris attending a workshop on operads. I learned a lot, and it was great to talk to Mathieu Anel, Steve Awodey, Benoit Fresse, Nicola Gambino, Ezra Getzler, Martin Hyland, André Joyal, Joachim Kock, Paul-André Melliès, Emily Riehl, Vladimir Voevodsky… and many other people to whom I apologize for not including in this prestigious list! (The great thing about senility is never having to say you’re sorry, but I haven’t quite reached that stage.)

There is a lot I could say… but that will have to wait for another time. For now I just want to point out this annotated video:

of a talk at the Catégories, Logiques, Etc… seminar at Paris 7, run by Anatole Khelif. This should be a fairly painless introduction to the subject, since I sensed that lots of people in the audience wanted me to start by explaining prerequisites: categorification, TQFTs, 2-Hilbert spaces and the Heisenberg algebra.

That means I didn’t manage to discuss other interesting things, like the definition of symmetric monoidal bicategory, or the role of combinatorics, especially Young diagrams. For those, go here and check out the links!

There are lots of other videos of talks on the website of Khelif’s seminar (all in French so far, except mine). For example, here are some on Olivia Caramello’s work on topos theory, and its relation to the Langlands program:

And finally, one more digression. I got invited to speak at this seminar thanks to the help of Andrée Ehresmann, whom I recently met at the Dagstuhl workshop Categories at the Crossroads. She also invited me to IRCAM, the big experimental music lab in Paris. I took a photo of her in an anechoic chamber:

If you’re interested in IRCAM or how Moreno Andreatta, Alexandre Popoff and Andrée Ehresmann are working on music theory with the help of categories, you can read a bit about it here.

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Re: Spans and the Categorified Heisenberg Algebra (Part 2)

I just got to look at the very nice slides for the talk with the same title as this post. From there I went to Mike Stay’s paper on the same subject, and I agree with John that the pictures are terrific!

The hexagonator diagrams are particularly interesting since they turn out to be the polytopes known as cyclohedra. (Here’s an encyclopedia entry that I’ve put together mentioning that fact.) Also in the encyclopedia entry are some links to other places that these polytopes crop up, in operad theory, graph theory and cluster algebras. Interestingly, the vertex labels from the operad point of view are close but not quite the same as the ones in Mike’s paper, from the viewpoint of braided monoidal bicategories. The vertices for the hexagon in operad theory would be labeled by (AB)C, A(BC), (BC)A, B(CA), (CA)B and C(AB). Compare those to the ones on page 15 of Mike’s paper. Mysterious.

Re: Spans and the Categorified Heisenberg Algebra (Part 2)

The appearance of this polytope is explained by it being the result of contracting edges in another polytope - the one labeled permutahedron in your link (a truncated octahedron) which is really just the graph of 2-morphisms on a 4-cube.
The 4-cube in question is the image of a 4-cube of finite sets when you view this whole system as a functor from finite sets.
The hexagons correspond to the sub 3-cubes and the squares correspond to the bicategory isomorphisms which “glue them together”.